Unitary engineering of two- and three-band Chern insulators

Dept. of physicsSungKyunKwan University, Korea

Soo-Yong Lee, Jin-Hong Park, Gyungchoon Go, Jung Hoon Han

arXiv:1312.6469[cond-mat]

APS Meeting, 03-03-2014, Denver

Contents

1. Introduction : Dirac monopole

2. Two-band Chern insulator 3. Three-band Chern insulator

4. Topological Band switching

5. Conclusion

Dirac MonopoleMaxwell eq. with magnetic monopole

Dirac quantization

Vector potential corresponding to monopole

Singularity at :

Dirac string

Ray, Nature (2014)

Fang, Science (2003)

Dirac MonopoleVector potential corresponding to wave function z

CP1 wave function corresponding to monopole vector potential

z is nothing but the spin coherent state of two-band spin Hamiltonian

Dirac monopole always appears in the general two-band spin model!!

Two-band Chern Insulator

Monopole charge = Chern number in Two-band Chern In-sulator

3-dim d-vector Pauli matrices

ex) Hall conductivity of the quantum Hall insulator

Two-band Chern InsulatorQ) How do we change the monopole charge?A) Unitary transformation !

Additional term by a certain unitary transformation can put in an extra sin-gular vector potential which generates a higher Chern number.

New Hamiltonian New wave ft.

New vector poten-tial

Two-band Chern Insulator

1) : Turning Chern number on and off.

2) : Increasing Chern number.

cf) Eigenvalues are always +d and –d and z is independent of the magnitude of d-vector

How to generate an arbitrary Chern number insulator?

1. Write down the unity Chern number model in the mo-mentum space. (ex : Haldane model, BHZ model etc.)

2. Apply the unitary transformation to change angle and 3. New d-vector gives a higher Chern number model4. (When we apply the Fourier transformation, we get a real space model. To avoid non-valid hopping, multi-orbital character could be sometimes introduced. ex) p-orbital, t2g-orbital

C=1 C=2Ex)

Three-band Chern Insulator

Gell-mann matrices

8-dim n-vector

Three-band Chern Insulator

Chern number of each band

SU(3) Euler rotation

c, d disappear in Hamiltonian

Gell-mann matrices

8-dim n-vector

Three-band Chern Insulator

A pair of monopole charges = Combination of the two band Chern insulartor (b=0)

Two redundant U(1) gauges c and d (non-degenerate)

1) : Increasing Chern number of one monopole.

2) : Increasing Chern number of another monopole

Band SwitchingAnother class of the three-band model

3-dim d-vector

Chern number of each band(factor 2 difference from the two-band model, He et al. PRB 2012 Go et al. PRB 2013)

Ex) Kagome lattice model(Ohgushi, Murakami, Nagaosa PRB 1999)

Spin-1 matricescf) Eigenvalues are always +d,0,-d

Band SwitchingBand switching by basis change unitary transformation

Ex)

3-dim Reminder of 8-dim except 3-dim d vector space (orthogonal)

Generally,

cf)

Band Switching

Ex) In the Kagome lattice, the additional term represents the next(n), nn, nnn hoppings through the center of the hexagon.

Edge state

Eigenenergies :

Jo et al, PRL (2012)

only when

unitary Trans.

Topological phase transition!

Conclusion

• Monopole charge-changing operations become unitary transformations on the two-band Hamiltonian.

• For the three-band case, we propose a topology-engi-neering scheme based on the manipulation of a pair of magnetic monopole charges.

• Band-switching is proposed as a way to control the topological ordering of the three-band Hamiltonian.

Thank you for your attention!